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$L^p$-estimates for the wave equation with partial inverse-square potentials

Jialu Wang, Chengbin Xu, Fang Zhang, Junyong Zhang

Abstract

This paper investigates $L^p$-estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schrödinger operator $\mathcal{H}_a = -Δ_x - Δ_y + a/|x|^2$ on $\mathbb{R}^{2+n}$. Using spectral analysis, we establish the $L^p$-boundedness of the wave propagator $(1+\sqrt{\mathcal{H}_a})^{-γ} e^{it\sqrt{\mathcal{H}_a}}$ for a range of exponents $γ$ and $p$ satisfying $|1/p -1/2| < γ/(n+1)$. The key ingredients are the spectral measure kernel of the partial inverse-square operator $\mathcal{H}_a$ and the complex interpolation argument.

$L^p$-estimates for the wave equation with partial inverse-square potentials

Abstract

This paper investigates -estimates for solutions to the wave equation perturbed by a scaling-critical partial inverse-square potential. We study a model in which the singularity of the potential appears only in a subset of the variables, corresponding to the Schrödinger operator on . Using spectral analysis, we establish the -boundedness of the wave propagator for a range of exponents and satisfying . The key ingredients are the spectral measure kernel of the partial inverse-square operator and the complex interpolation argument.

Paper Structure

This paper contains 5 sections, 6 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Proposition \ref{['prop:tz_bounds']}
  4. Proof of Theorem \ref{['thm:main']}
  5. Integrability of Singular Integral Kernels

Key Result

Theorem 1.1

Let $\mathcal{H}_{a}$ be in La and let $\gamma > 0$ and $1 \leq p \leq \infty$ satisfy $|\frac{1}{p} - \frac{1}{2}| < \frac{\gamma}{n+1}$. Then there exists a constant $C(p, \gamma) > 0$ such that for any $t > 0$ and all $f \in L^{p}(\mathbb{R}^{2+n}),$ we have

Theorems & Definitions (10)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.1
  • Proposition 2.2
  • Definition 2.1
  • Proposition 3.1
  • proof
  • proof
  • Lemma A.1
  • proof